Theorem efrunt

Description: If A is well-founded by _E , then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011)

	Ref	Expression
Assertion	efrunt	⊢ ( E Fr 𝐴 → ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		frirr	⊢ ( ( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑥 E 𝑥 )
2		epel	⊢ ( 𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥 )
3	1 2	sylnib	⊢ ( ( E Fr 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑥 ∈ 𝑥 )
4	3	ralrimiva	⊢ ( E Fr 𝐴 → ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 )